for which the expression of \(\ell _0\)  can be found in [1]}, and only contributions from up to the second harmonics
are included here because [1]} did not provide expressions for contributions from higher harmonics.
The phase function is given by \(\theta = k_0 x - \Omega _0 t\) .
The “classical” reconstruction procedure based on (REF ) clearly differs from the present approach.
It is more explicit and thus computationally more efficient but is perturbative.
Contributions at each order up to the desired one need to be derived and their expressions become increasingly complicated.
On the other hand, our Hamiltonian procedure requires solving an auxiliary system of PDEs
to reconstruct \(\eta \)  and \(\zeta \)  (or \(\xi \) ) from \(u\)  but it is non-perturbative.
Indeed, Eqs. (REF ) and (REF )
constitute an exact representation of the Birkhoff normal form transformation that eliminates non-resonant triads in this problem.
